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Next: Conclusions Up: Conserved symmetries Previous: Proton-neutron symmetry


Time-reversal symmetry

In the case of time-reversal invariance, $\hat{\rho}^T$=$\hat{\rho}$ and $\hat{\breve{\rho}}^T$= $\hat{\breve{\rho}}$, see Eqs. (7), the p-h and p-p densities fulfill additional conditions,

$\displaystyle {\rho}_0(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ $\textstyle =$ $\displaystyle {\rho}^*_0(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}'), \quad$ (217)
$\displaystyle {\rho}_k(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ $\textstyle =$ $\displaystyle -(-1)^{k}{\rho}^*_k(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}'),$ (218)
$\displaystyle {\mbox{{\boldmath {$s$}}}}_0(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ $\textstyle =$ $\displaystyle -{\mbox{{\boldmath {$s$}}}}^*_0(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}'), \quad$ (219)
$\displaystyle {\mbox{{\boldmath {$s$}}}}_k(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ $\textstyle =$ $\displaystyle (-1)^{k}{\mbox{{\boldmath {$s$}}}}^*_k(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ (220)

and
$\displaystyle \breve{\rho}_0(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ $\textstyle =$ $\displaystyle \breve{\rho}^*_0(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}'), \quad$ (221)
$\displaystyle \breve{\rho}_k(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ $\textstyle =$ $\displaystyle -(-1)^{k}\breve{\rho}^*_k(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}'),$ (222)
$\displaystyle \breve{\mbox{{\boldmath {$s$}}}}_0(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ $\textstyle =$ $\displaystyle -\breve{\mbox{{\boldmath {$s$}}}}^*_0(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}'), \quad$ (223)
$\displaystyle \breve{\mbox{{\boldmath {$s$}}}}_k(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ $\textstyle =$ $\displaystyle (-1)^{k}\breve{\mbox{{\boldmath {$s$}}}}^*_k(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}'),$ (224)

where $k$=1,2,3. Due to the fact that the $k$=2 Pauli matrix $\hat{\tau}^2$ is imaginary, the time reversal acts differently on the $k$=2 isovector components than on the $k$=1,3 components of all isovector densities. At the first sight, this seems to be a bizarre property. Indeed, the isospin quantum number is introduced to take into account the fact that there are two kinds of nucleons in nature, and each kind has its own, apparently unrelated to one another, time-reversal operation.

However, the use of the standard isospin formalism implies something more. Namely, the neutron wave function (isospin up) can be obtained from the proton wave function (isospin down) by an action of the (real) $\hat{\tau}^1$ Pauli matrix. Therefore, the relative phases of the neutron and proton wave functions are fixed by the phase convention that has been used to choose the isospin Pauli matrices. As a consequence, the time-reversal properties of neutrons and protons are not any more independent from one another. Of course, this is not a spurious quirk of the mathematics we use, but a reflection of a deeper fact that by mixing the neutron and proton wave functions we introduce complex mixing coefficients that do affect the time-reversal properties of the mixed wave function. Conservation of the time reversal means that these mixing coefficients must follow rules dictated by the time reversal, which implies differences between the $k$=2 and $k$=1,3 iso-directions. Therefore, we see here that from basic arguments it follows that conservation of the time reversal must imply the isospin symmetry breaking. The only iso-rotations that are compatible with the time reversal are those about the $k$=2 iso-axis. (The influence of the time-odd fields on the magnitude of the Wigner energy was pointed out in Ref. [12].)

Table 3 summarizes properties of p-h and p-p densities under the exchange of their spatial arguments. When no conserved symmetry is imposed, all densities are complex, and their real and imaginary parts are either symmetric or antisymmetric. For conserved time reversal, all densities become either real or imaginary, and are either symmetric or antisymmetric. Recall that symmetric parts contribute only to particle, kinetic, spin, spin-kinetic, and tensor-kinetic local densities, while the antisymmetric parts contribute only to the current and spin-current local densities. Therefore, local densities are complex, real, imaginary, or vanishing, depending on whether time-reversal, proton-neutron, or both symmetries are conserved. Table 4 presents these properties for all local p-h and p-p densities.

In previous studies, e.g., in Refs. [152,153,3,62,65], the $T$=1 pairing fields were associated with the real part of the pairing tensor, while the $T$=0 pairing was represented by the imaginary part of the pairing tensor. Such a structure was obtained for specific phase conventions and symmetries. On the other hand, as shown in Table 4, the general case corresponding to no conserved symmetries (e.g., for rotating states) requires that all the pn densities be complex.


Table 3: Symmetries of the p-h (left) and p-p (right) densities in general case (no conserved symmetries imposed), and in case of the time-reversal symmetry conserved. Real ($\Re$) and imaginary ($\Im$) parts are symmetric (S) or antisymmetric (A) under exchange of their spatial arguments, as indicated in the Table.
  general time-rev.       general time-rev.
density $\Re$ $\Im$ $\Re$ $\Im$     density $\Re$ $\Im$ $\Re$ $\Im$
$ \rho_0 (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ S A S 0      $\breve{\rho}_0 (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ A A A 0
$ \rho_2 (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ S A 0 A      $\breve{\rho}_2 (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ S S 0 S
$ \rho_{1,3} (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ S A S 0      $\breve{\rho}_{1,3} (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ S S S 0
$ \mbox{{\boldmath {$s$}}}_0 (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ S A 0 A      $\breve{\mbox{{\boldmath {$s$}}}}_0 (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ S S 0 S
$ \mbox{{\boldmath {$s$}}}_2 (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ S A S 0      $\breve{\mbox{{\boldmath {$s$}}}}_2 (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ A A A 0
$ \mbox{{\boldmath {$s$}}}_{1,3}(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ S A 0 A      $\breve{\mbox{{\boldmath {$s$}}}}_{1,3}(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ A A 0 A


Table 4: Properties of the local p-h and p-p densities in general case (no conserved symmetries imposed), and in case of the time-reversal, proton-neutron, or both symmetries conserved. The $k$=0,1,2, or 3 isospin components of densities are complex (C), real (R), imaginary (I), or zero (0), as indicated in the Table.
  general time-rev. prot.-neut. both
k 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3
$ \rho_k$ R R R R R R 0 R R 0 0 R R 0 0 R
$ \tau_k$ R R R R R R 0 R R 0 0 R R 0 0 R
$ {\mathsf J}_k$ R R R R R R 0 R R 0 0 R R 0 0 R
$ \mbox{{\boldmath {$s$}}}_k$ R R R R 0 0 R 0 R 0 0 R 0 0 0 0
$ \mbox{{\boldmath {$T$}}}_k$ R R R R 0 0 R 0 R 0 0 R 0 0 0 0
$ \mbox{{\boldmath {$j$}}}_k$ R R R R 0 0 R 0 R 0 0 R 0 0 0 0
$ \mbox{{\boldmath {$F$}}}_k$ R R R R 0 0 R 0 R 0 0 R 0 0 0 0
$\breve{\rho}_k$ 0 C C C 0 R I R 0 C C 0 0 R I 0
$\breve{\tau}_k$ 0 C C C 0 R I R 0 C C 0 0 R I 0
$\breve{{\mathsf J}}_k$ 0 C C C 0 R I R 0 C C 0 0 R I 0
$\breve{\mbox{{\boldmath {$s$}}}}_k$ C 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0
$\breve{\mbox{{\boldmath {$T$}}}}_k$ C 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0
$\breve{\mbox{{\boldmath {$j$}}}}_k$ C 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0
$\breve{\mbox{{\boldmath {$F$}}}}_k$ C 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0


To summarize this subsection, we now enumerate all non-zero densities when the time reversal is conserved or not, and/or the proton-neutron mixing is present or not. By counting as one density each component of a vector, tensor, or isovector, we obtain the following four options:



$1^\circ$ Time reversal broken plus proton-neutron mixing:

-
23 real p-h isoscalar densities: $\rho_0(\mbox{{\boldmath {$r$}}})$, $\tau_0(\mbox{{\boldmath {$r$}}})$, ${\mathsf J}_0(\mbox{{\boldmath {$r$}}})$, $\mbox{{\boldmath {$s$}}}_0(\mbox{{\boldmath {$r$}}})$, $\mbox{{\boldmath {$T$}}}_0(\mbox{{\boldmath {$r$}}})$, $\mbox{{\boldmath {$j$}}}_0(\mbox{{\boldmath {$r$}}})$, and $\mbox{{\boldmath {$F$}}}_0(\mbox{{\boldmath {$r$}}})$,
-
69 real p-h isovector densities: $\vec{\rho}(\mbox{{\boldmath {$r$}}})$, $\vec{\tau}(\mbox{{\boldmath {$r$}}})$, $\vec{{\mathsf J}}(\mbox{{\boldmath {$r$}}})$, $\vec{\mbox{{\boldmath {$s$}}}}(\mbox{{\boldmath {$r$}}})$, $\vec{\mbox{{\boldmath {$T$}}}}(\mbox{{\boldmath {$r$}}})$, $\vec{\mbox{{\boldmath {$j$}}}}(\mbox{{\boldmath {$r$}}})$, and $\vec{\mbox{{\boldmath {$F$}}}}(\mbox{{\boldmath {$r$}}})$,
-
12 complex p-p isoscalar densities: $ \breve{\mbox{{\boldmath {$s$}}}}_0(\mbox{{\boldmath {$r$}}})$, $ \breve{\mbox{{\boldmath {$T$}}}}_0(\mbox{{\boldmath {$r$}}})$, $ \breve{\mbox{{\boldmath {$j$}}}}_0(\mbox{{\boldmath {$r$}}})$, and $ \breve{\mbox{{\boldmath {$F$}}}}_0(\mbox{{\boldmath {$r$}}})$,
-
33 complex p-p isovector densities: $\vec{\breve {\rho}}(\mbox{{\boldmath {$r$}}})$, $\vec{\breve {\tau}}(\mbox{{\boldmath {$r$}}})$, and $\vec{\breve{\mathsf J}}(\mbox{{\boldmath {$r$}}})$,

$2^\circ$ Time reversal conserved plus proton-neutron mixing:

-
11 real p-h isoscalar densities: $\rho_0(\mbox{{\boldmath {$r$}}})$, $\tau_0(\mbox{{\boldmath {$r$}}})$, and ${\mathsf J}_0(\mbox{{\boldmath {$r$}}})$,
-
30 real p-h isovector densities: $ {\rho}_{1,3}(\mbox{{\boldmath {$r$}}})$, $ {\tau}_{1,3}(\mbox{{\boldmath {$r$}}})$, $ {{\mathsf J}}_{1,3}(\mbox{{\boldmath {$r$}}})$, $ {\mbox{{\boldmath {$s$}}}}_2 (\mbox{{\boldmath {$r$}}})$, $ {\mbox{{\boldmath {$T$}}}}_2 (\mbox{{\boldmath {$r$}}})$, $ {\mbox{{\boldmath {$j$}}}}_2 (\mbox{{\boldmath {$r$}}})$, and $ {{\mathsf J}}_2 (\mbox{{\boldmath {$r$}}})$,
-
12 imaginary p-p isoscalar densities: $ \breve{\mbox{{\boldmath {$s$}}}}_0(\mbox{{\boldmath {$r$}}})$, $ \breve{\mbox{{\boldmath {$T$}}}}_0(\mbox{{\boldmath {$r$}}})$, $ \breve{\mbox{{\boldmath {$j$}}}}_0(\mbox{{\boldmath {$r$}}})$, and $ \breve{\mbox{{\boldmath {$F$}}}}_0(\mbox{{\boldmath {$r$}}})$,
-
33 p-p isovector densities, 22 real: $\breve {\rho}_{1,3}(\mbox{{\boldmath {$r$}}})$, $\breve {\tau}_{1,3}(\mbox{{\boldmath {$r$}}})$, $\breve{\mathsf J}_{1,3}(\mbox{{\boldmath {$r$}}})$, and 11 imaginary: $\breve {\rho}_2 (\mbox{{\boldmath {$r$}}})$, $\breve {\tau}_2 (\mbox{{\boldmath {$r$}}})$, $\breve{\mathsf J}_2 (\mbox{{\boldmath {$r$}}})$,

$3^\circ$ Time reversal broken, no proton-neutron mixing:

-
23 real p-h isoscalar densities: $\rho_0(\mbox{{\boldmath {$r$}}})$, $\tau_0(\mbox{{\boldmath {$r$}}})$, ${\mathsf J}_0(\mbox{{\boldmath {$r$}}})$, $\mbox{{\boldmath {$s$}}}_0(\mbox{{\boldmath {$r$}}})$, $\mbox{{\boldmath {$T$}}}_0(\mbox{{\boldmath {$r$}}})$, $\mbox{{\boldmath {$j$}}}_0(\mbox{{\boldmath {$r$}}})$, and $\mbox{{\boldmath {$F$}}}_0(\mbox{{\boldmath {$r$}}})$,
-
23 real p-h isovector densities: $ {\rho}_3(\mbox{{\boldmath {$r$}}})$, $ {\tau}_3(\mbox{{\boldmath {$r$}}})$, $ {{\mathsf J}}_3(\mbox{{\boldmath {$r$}}})$, $ {\mbox{{\boldmath {$s$}}}}_3(\mbox{{\boldmath {$r$}}})$, $ {\mbox{{\boldmath {$T$}}}}_3(\mbox{{\boldmath {$r$}}})$, $ {\mbox{{\boldmath {$j$}}}}_3(\mbox{{\boldmath {$r$}}})$, and $ {\mbox{{\boldmath {$F$}}}}_3(\mbox{{\boldmath {$r$}}})$,
-
22 complex p-p isovector densities: $\breve {\rho}_{1,2}(\mbox{{\boldmath {$r$}}})$, $\breve {\tau}_{1,2}(\mbox{{\boldmath {$r$}}})$, and $\breve{\mathsf J}_{1,2}(\mbox{{\boldmath {$r$}}})$,

$4^\circ$ Time reversal conserved, no proton-neutron mixing:

-
11 real p-h isoscalar densities: $\rho_0(\mbox{{\boldmath {$r$}}})$, $\tau_0(\mbox{{\boldmath {$r$}}})$, and ${\mathsf J}_0(\mbox{{\boldmath {$r$}}})$,
-
11 real p-h isovector densities: $ {\rho}_3(\mbox{{\boldmath {$r$}}})$, $ {\tau}_3(\mbox{{\boldmath {$r$}}})$, and $ {{\mathsf J}}_3(\mbox{{\boldmath {$r$}}})$,
-
22 p-p isovector densities, 11 real ${\breve {\rho}}_1(\mbox{{\boldmath {$r$}}})$, ${\breve {\tau}}_1(\mbox{{\boldmath {$r$}}})$, ${\breve{\mathsf J}}_1(\mbox{{\boldmath {$r$}}})$, and 11 imaginary ${\breve {\rho}}_2(\mbox{{\boldmath {$r$}}})$, ${\breve {\tau}}_2(\mbox{{\boldmath {$r$}}})$, ${\breve{\mathsf J}}_2(\mbox{{\boldmath {$r$}}})$.

next up previous
Next: Conclusions Up: Conserved symmetries Previous: Proton-neutron symmetry
Jacek Dobaczewski 2004-01-03